Local and multilinear noncommutative de Leeuw theorems

Let $\Gamma<G$ be a discrete subgroup of a locally compact unimodular group $G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p<\infty$ and let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding Fourier multiplier. Similarly, let $T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma})$ be the Fourier multiplier associated to the restriction $m|_{\Gamma}$ of $m$ to $\Gamma$. We show that $$ c( {\rm supp}( m|_{\Gamma} ) ) \Vert T_{m \vert_\Gamma}: L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma}) \Vert \leq \Vert T_{m }: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, $$ for a specific constant $0 \leq c(U) \leq 1$ that is defined for every $U \subseteq \Gamma$. The function $c$ quantifies the failure of $G$ to admit small almost $\Gamma$-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, $c(\Gamma) =1$ when $G$ has small almost $\Gamma$-invariant neighbourhoods. Our result thus extends the De Leeuw restriction theorem from [CPPR15] as well as De Leeuw's classical theorem [Lee65]. For real reductive Lie groups $G$ we provide an explicit lower bound for $c$ in terms of the maximal dimension $d$ of a nilpotent orbit in the adjoint representation. We show that $c(B_\rho^G) \geq \rho^{-d/2}$ where $B_\rho^G$ is the ball of $g\in G$ with $\Vert {\rm Ad}_g \Vert<\rho$. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear De Leeuw restriction theorem for pairs $\Gamma<G$ with $c(\Gamma) = 1$. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.


Introduction
In his seminal paper Karel de Leeuw [Lee65] proved several fundamental theorems about Fourier multipliers on the Euclidean space R n . Recall that m ∈ L ∞ (R n ) is called a p-multiplier if the map T m on L 2 (R n ) determined by F 2 (T m f ) = mF 2 (f ) with F 2 the unitary Fourier transform extends to a bounded map on L p (R n ). Using Pontrjagin duality the definition of p-multipliers may be extended to any locally compact abelian group G and we call m ∈ L ∞ (G) a p-multiplier if the map T m on L 2 ( G) determined by F 2 (T m f ) = mF 2 (f ) extends to a bounded map on L p ( G). Recall here that in case of R n we have that L p ( R n ) is isomorphic to L p (R n ) canonically by conjugating with F 2 .
One of De Leeuw's most important contributions is the restriction theorem. Let H < R n be any subgroup equipped with the discrete topology. If m ∈ C b (R n ) is a p-multiplier then m| H ∈ C b (H) is a p-multiplier. Moreover, A particular case occurs when H = R n disc is R n equipped with the discrete topology. Then R n disc is the Bohr compactification of R n . This special case is usually referred to as the compactification theorem and is essentially the strongest theorem proved in [Lee65]. Its proof consists of first showing (1.1) when H is discrete in the Euclidean topology inherited from R n combined with a lattice approximation theorem. These theorems became standard tools in harmonic analysis and are the first instances of so-called transference results.
The emergence of noncommutative integration has created a realm in which Fourier multipliers can be interpreted naturally for any locally compact group. Here function algebras on the Pontrjagin dual group are replaced by group algebras and their C * -and von Neumann closures. The theory of noncommutative L p -spaces [Nel74], [PiXu03] and realization of Fourier transforms [Kun58], [Coo10], [Cas13] leads to a natural notion of p-multipliers. In this interpretation G plays the role of the frequency side and for m ∈ L ∞ (G) the Fourier multiplier T m acts on a noncommutative space, namely the noncommutative L p -space of the group von Neumann algebra of G.
For p = ∞ the relevance of Fourier multipliers on group von Neumann algebras was already recognized in the fundamental results of Haagerup on approximation properties [Haa78], as well as the works by De Cannière-Haagerup [CaHa85] and Cowling-Haagerup [CoHa89] on approximation properties of real rank one simple Lie groups.
There is a significant interest in understanding the class of p-multipliers beyond p = ∞. Already in commutative analysis this has triggered an enormous machinery of harmonic analysis including the study of singular integrals, Calderón-Zygmund theory, Hörmander-Mikhlin multipliers, et cetera, see e.g. [Gra08], [Gra09], [Ste70]. The study of multipliers is further motivated by their applications to convergence properties of Fourier series, the structure of Banach spaces and the analysis of pseudo-differential operators.
In the noncommutative situation several efforts have been made in the understanding of pmultipliers. In [JMP14] multipliers associated with cocycles on locally compact groups were constructed; such multipliers are compositions m = m • b with b : G → H a cocycle into a finite dimensional orthogonal representation (H, π) with additional Hörmander-Mikhlin conditions on m. Further results in this direction have been obtained in [JMP18], [JGP17]. Note that if G has property (T) then a well-known theorem by Delorme-Guichardet [Del77], [Gui72] (see [BHV08, Sections 2.2, 2.12]) shows that every cocycle b is bounded and therefore the multiplier m = m • b has a certain oscillatory behaviour. Note that if G = SL(n, R) the lack of nontrivial orthogonal representations limits this method. However, concrete multipliers on SL(n, R) have been constructed in [PRS] where a version of the Hörmander-Mikhlin multiplier theorem was obtained for the Lie group SL(n, R), the real n × n-matrices with determinant 1. Here the Hörmander-Mikhlin conditions are formulated in the natural differentiable structure of the Lie group. In the discrete setting Cotlar's identity has successfully been applied [MeRi17], [MRX] to obtain Hilbert transforms and Hörmander-Mikhlin multipliers on free groups.
It is natural to ask which relations there are between p-multipliers on G and p-multipliers on the discrete subgroups of G. A cornerstone theorem was obtained in [CPPR15] where the De Leeuw restriction theorem was obtained for any pair Γ < G with Γ a discrete subgroup of a (for simplicity unimodular) locally compact group G such that G has small almost Γ-invariant neighbourhoods (see Definition 3.1). If Γ is amenable then G always has small almost Γ-invariant neighbourhoods [CPPR15,Theorem 8.7]. In particular the theorem is applicable to nilpotent locally compact groups.
There are several deep open problems that concern the relation of multipliers to their restrictions on subgroups beyond the cases covered by [CPPR15]. For instance, the existence of explicit p-multipliers on SL(n, Z), n ≥ 3 remains an open problem. In particular, there does not seem to be a genuine example of a p-multiplier with compact support or with sufficient decay. This paper does not directly solve this problem but it provides a potential entry as it would be sufficient to construct multipliers with sufficient bounds on SL(n, R); this also closely relates to the Calderón-Torchinsky problem mentioned in [PRS,Section 5.C]. Other questions are raised by the approximation and rigidity properties of noncommutative L p -spaces obtained in [JuRu03], [Laf11], [Laa13], [LaSa18], [PRS, Theorem B].
Another direction that is initiated in this paper is the theory of multilinear Fourier multipliers on noncommutative L p -spaces. In commutative harmonic analysis this became a large topic including milestone results by Lacey-Thiele on the bilinear Hilbert transform [LaTh99] and the development of multilinear Calderón-Zygmund theory [GrTo02]. Also very recently, several results in the semicommutative/vector-valued setting have been obtained [AmUr20], [DLMV20a], [DLMV20b]. In the noncommutative setting multilinear Schur multipliers and operator integrals have already provided very deep applications such as the resolution of Koplienko's conjecture [PSS13]. The theory of Fourier multipliers -which by transference is intimately related to Schur multipliers [Neu11], [CaSa15] -seems to be undeveloped beyond abelian groups in the multilinear setting. We shall prove the multilinear versions of De Leeuw's restriction, lattice approximation and compactification theorem. These were proved in the abelian (bilinear) setting in [Bla05], [Rod13].
We now summarize the main results of this paper. We shall introduce a quantified version of having small almost invariant neighbourhoods. More precisely, we consider the following definition.
Definition 1.1. Let G be a locally compact group with (left) Haar measure µ. Let F ⊆ G be arbitrary and let V ⊆ G be relatively compact of non-zero measure. We introduce the quantity For a neighbourhood basis V of the identity of G we set δ F (V) = lim inf V ∈V δ F (V ). Then set δ F to be the supremum of δ F (V) over all symmetric neighbourhood bases V of the identity.
We prove the following local version of the noncommutative De Leeuw restriction theorem [CPPR15, Theorem A]. We call this theorem 'local' since we obtain an estimate that is controlled by a subset U of Γ and which holds for all symbols m ∈ C b (G) such that the restriction m| Γ is supported on U .
From this point let Γ be a discrete subgroup of any locally compact unimodular group G.
Theorem A. Let m ∈ C b (G). Then for every 1 ≤ p < ∞ we have that where c(U ) = inf{δ 1 2 F | F ⊆ U, F finite} is defined for every U ⊆ Γ. Theorem A holds as well if the bounds of both multipliers are replaced by their complete bounds (see proof of Proposition 3.16). The strength of Theorem A is that the constant c can be determined explicitly in many interesting situatons. We have c(Γ) = 1 if and only if G has small almost Γ-invariant neighbourhoods. Hence, our theorem recovers [CPPR15,Theorem A]. In Sections 8, 9 and 10 we find very natural lower estimates on δ F in the full generality of real reductive Lie groups. For ρ ≥ 1 let B G ρ = {g ∈ G | Ad g ≤ ρ} where Ad is the adjoint representation.
Theorem B. Let G be a real reductive Lie group. For ρ > 1 we have δ B G ρ ≥ ρ −d/2 where d is the maximal dimension of a nilpotent adjoint orbit.
In order to prove Theorem B we first reduce it to the connected adjoint group of G, which is semisimple. We then construct a symmetric neighbourhood basis V g ε,R , ε > 0, R > 0 of 0 in the Lie algebra g such that This implies Theorem B. The neighbourhood basis is constructed in such a way that ε>0 V g ε,R is the intersection of the nilpotent cone N ⊆ g with the ball of radius R around the origin. Note that the union of the nilpotent orbits O X of maximal dimension d is dense in N . Using results of Harisch-Chandra and Barbasch-Vogan on limiting orbit integrals, we show that the scaling behaviour of µ(exp(V g ε,R )) is governed by the scaling behaviour of the Liouville form of O X . Since the KKS symplectic form ω KKS scales as ρ under dilation, the Liouville form 1 (d/2)! ∧ d/2 ω KKS scales as ρ d/2 .
In Example 3.14 we consider the concrete case Γ = SL(n, Z), G = SL(n, R). We discuss how Theorems A and B give an ansatz to construct multipliers on Γ = SL(n, Z).
Next we start the analysis of multilinear multipliers. Most efforts are required for the restriction theorem.
Theorem C. Let G be a locally compact unimodular group. Let Γ < G be a discrete subgroup such that c(Γ) = 1. Let m ∈ C b (G ×n ). Then for every 1 ≤ p, p 1 , . . . , p n < ∞ with p −1 = n j=1 p −1 j we have that Theorem C hinges on the intertwining property of Lemma 4.6 whose proof is very delicate and which requires several new ideas compared to its linear counterpart. The global idea is to approximate m with multipliers m k that can be expressed in terms of nested compositions of linear ∞-multipliers. We then construct asymptotically isometric maps that intertwine the Fourier multipliers associated with m k | Γ ×n and m k . The condition c(Γ) = 1 is used at several places in the proof, whereas in the linear case it is only needed to construct asymptotically isometric maps. This also explains why we do not obtain a 'local' version of Theorem C as well.
We further obtain multilinear versions of the lattice approximation theorem (Theorem 5.2) as well as the multilinear compactification theorem and the periodization theorem (Section 6). We use these theorems to construct examples of multilinear Fourier multipliers on the Heisenberg group. These are the first genuine examples of multilinear Fourier multipliers on a nonabelian group.
Structure of the paper. Section 2 contains preliminaries on noncommutative L p -spaces of group von Neumann algebras. Section 3 proves Theorem A. We also establish approximate embeddings of the non-commutative L p -spaces assocatied with an inclusion of groups Γ < G. Section 4 defines multilinear Fourier multipliers and proves Theorem C. The proofs of the remaing De Leeuw theorems are contained in Sections 5 and 6; this concerns the lattice approximation, compactification and periodization. Section 7 contains examples on the Heisenberg group. In Sections 8, 9, 10 we prove Theorem B; we have postponed the proof of Theorem B to the end of the paper so that all Lie theoretic arguments are presented separately.
Acknowledgement. BJ and LM are grateful to Tobias Diez for interesting discussions and for providing crucial references. MC wishes to thank Adrián González-Pérez, Javier Parcet andÉric Ricard for useful discussions and communication.
2. Non-commutative L p -spaces of group von Neumann algebras N denotes the natural numbers starting from 1. We denote N ≥0 = N ∪ {0}.
2.1. Assumptions on groups. All groups G in this paper are assumed to be locally compact, second countable and unimodular. Though our results can be stated without the second countability assumption, it significantly simplifies the exposition of our proofs as we can work with neighbourhood bases of the identity instead of shrinking nets of neighbourhoods. When we say that Γ is a discrete subgroup of G we mean that it is discrete in the topology of G. Any discrete subgroup Γ < G is then countable.
For s, t ∈ G we write Ad s (t) = sts −1 . We denote µ for the Haar measure on G. We use the shorthand notation ds = dµ(s) for integrals with respect to µ. A set V ⊆ G is called symmetric if V = V −1 . A neighbourhood basis of the identity is called symmetric if it consists of symmetric sets.
2.2. Von Neumann algebras. A von Neumann algebra M is a unital * -subalgebra of bounded operators on a Hilbert space that is closed in the strong topology. M is called semi-finite if it admits a faithful normal trace τ : M + → [0, ∞]. M is finite in case there exists such τ with τ (1) = 1, i.e. τ extends to a state on M .For general von Neumann algebra theory and noncommutative integration we refer to [Tak02], [StZs79].
2.3. Group von Neumann algebras. Let C b (G) be the continuous bounded functions G → C and let C c (G) be the subspace of compactly supported functions. Let L p (G), 1 ≤ p < ∞ be the Banach space of functions G → C that are p-integrable, meaning f p : We further set be the left regular representation. Set λ(f ) = G f (s)λ(s)d(s) where the integral is convergent in the strong topology. We have We set the group von Neumann algebra where span denotes the strong closure of the linear span. There exists a unique normal semi-finite faithful weight ϕ G on L(G) that is defined as follows. For x ∈ L(G) we have otherwise.
ϕ G is tracial, meaning that for all x ∈ L(G), ϕ G (x * x) = ϕ G (xx * ), iff G is unimodular, which we will always assume. ϕ G is a state if and only if G is discrete.
2.4. Crossed products. Let M be a semi-finite von Neumann algebra with a trace τ M and let Γ be a discrete group acting on M ⊆ B(H) via a trace preserving action θ : Γ → Aut(M ). For This has a natural tracial weight τ which extends τ M on {ι(x) : x ∈ M } and τ Γ on 1 ⊗ λ(g) : g ∈ Γ (c.f. [Tak02]).
2.5. Non-commutative L p -spaces. The results in this section can be found in [Kun58], [Nel74], [Ter81], [PiXu03], [Cas13]. For an exponent p ∈ [1, ∞] we will write p ′ ∈ [1, ∞] for the conjugate exponent set by 1 p + 1 p ′ = 1. Let M be a von Neumann algebra equipped with a normal, semi-finite, faithful trace τ . For 1 ≤ p < ∞, set We define L p (M, τ ) as the completion of {x ∈ M | x p < ∞} with respect to · p . Alternatively, L p (M, τ ) may be described as the space of all closed, densely defined operators x that are affiliated with M and for which x p p := τ (|x| p ) < ∞. The latter description is more concrete but requires the introduction of affiliated operators and extension of the trace thereon; this shall not be used further in this paper. We define L ∞ (M, τ ) to be M with the operator norm. Then L p (M, τ ), 1 ≤ p ≤ ∞ is a Banach space and we have a Hölder inequality The Hölder inequality is moreover sharp in the sense that indeed if x = u|x| is the polar decomposition of x then the supremum is attained at y = |x| p q . The trace τ may be extended from M ∩ L 1 (M, τ ) to L 1 (M, τ ) linearly and continuously in · 1 . For x ∈ L p (M, τ ) and y ∈ L p ′ (M, τ ) we have in particular that yx ∈ L 1 (M, τ ) and we have a pairing (2.4) y, x p ′ ,p = τ (yx).
We have x p = x * p for any x ∈ L p (M, τ ). In case (M, τ ) is (L(G), ϕ G ) with G a unimodular locally compact group -so that ϕ G is tracial -we simply write L p ( G) for L p (L(G), ϕ G ). If G is abelian the latter space is isomorphic to the usual L p -space of the Pontrjagin dual group G. For 2 ≤ p ≤ ∞ and f ∈ C c (G) we have λ(f ) ∈ L p ( G) with λ(f ) p ′ ≤ f p (see [Kun58], [Coo10], [Cas13]) and such elements are dense. We denote C c (G) * 2 = C c (G) * C c (G) for the second convolution power of C c (G). The elements λ(f ), f ∈ C c (G) * 2 are dense in L p ( G), 1 ≤ p < ∞. The same densities hold for p = ∞ but then in the σ-weak topology of L ∞ ( G) = L(G). In each case the frequency support of λ(f ) is by definition the support of f . For φ, f ∈ λ(C c (G) * 2 ) we have It follows directly from (2.2) that the Plancherel identity holds: In case G = Γ is discrete the elements with finite frequency support are given by the group algebra which is dense in L p ( Γ) for any 1 ≤ p < ∞. Here c 00 (Γ) denotes the finitely supported functions Γ → C.
Remark 2.1. There are notions of non-commutative L p associated with an arbitrary von Neumann algebra due to Haagerup [Ter81] and Connes-Hilsum [Hil81]. This allows for the study of de Leeuw theorems for non-unimodular groups, see for instance [CPPR15,Section 8]. However, to keep the current paper broadly accessible we will work within the realm of unimodular groups and tracial von Neumann algebras.
2.6. Fourier multipliers. We say that a function m In particular this encompasses that λ(mf ) ∈ L q ( G). We briefly say p-multiplier in case p = q. By the Plancherel identity the space of 2-multipliers with continuous symbol is C b (G).
Remark 2.2. Let 2 ≤ p < ∞. Suppose that m ∈ C b (G) is a p-multiplier. Then under the pairing (2.5) the dual of T m : L p ( G) → L p ( G) is the p ′ -multiplier given by T m ∨ : L p ′ ( G) → L p ′ ( G).

Theorem A: The local linear De Leeuw restriction theorem
The aim of this section is to prove Theorem A. We also introduce the quantified version of local almost invariant neighbourhoods in Definition 1.1 which we shall determine for real reductive Lie groups in Section 8. We fix again a locally compact unimodular group G which we assume to be second countable. Let Γ < G be a discrete subgroup. Recall that µ is the Haar measure of G.
At this stage recall that δ F as was defined in Definition 1.1.  Remark 3.3. G has small invariant neighbourhoods (notation G ∈ [SIN] Γ ) with respect to Γ if there exists a neighbourhood basis V of the identity of G such that Ad s (V ) = V, s ∈ Γ.
Clearly then G ∈ [SAIN] Γ . Indeed, we may replace V ∈ V by V ∩ V −1 to obtain a symmetric neighbourhood basis with the desired properties.
Remark 3.4. Suppose that G ∈ [SAIN] Γ . Since G is first countable pick any neighbourhood basis (U i ) i∈N of the identity of G. Since G is second countable, Γ is countable. Let F i ⊆ Γ, i ∈ N be an increasing sequence of finite subsets whose union is Γ. For every i ∈ N we may inductively pick a relatively compact open neighbourhood V i of the identity in G such that 1 − 1 This construction has the advantage that we may fix a single neighbourhood basis V independent of the finite set F .
3.1. Asymptotic embeddings for 2 ≤ p < ∞. The aim of this section is to construct contractive maps L p ( Γ) → L p ( G) that are asymptotic embeddings. In case G ∈ [SAIN] Γ these maps are asymptotically isometric. Our results therefore generalize [CPPR15, Claim A].
is positive definite. Moreover A ≥ δ F (V )I where I = (1) s,t∈F is the matrix with all entries equal to 1.
Proof. Let V 0 := s∈F Ad s (V ) and define ξ : As usual we view C[Γ] ⊆ L(Γ) ⊆ L(G) as subalgebras of each other naturally.
Proposition 3.6. Let x ∈ C[Γ] with finite frequency support F ⊆ Γ. Let V be a relatively compact symmetric neighbourhood of the identity of G and assume that sV, s ∈ F are disjoint.
where we use the notation Φ ∞,V (x) = x. Then for every 2 ≤ p ≤ ∞ we have Proof. Write x = s∈F x s λ(s). By Remark 3.7 below and the Plancherel identity Since sV, s ∈ F are disjoint and using once more the Plancherel identity, This gives the upper estimate (3.1) for p = 2. For p = ∞ the estimate (3.1) is trivial. From the three lines lemma (similar to Stein's interpolation theorem for analytic families of maps [Ste56]) we then have (3.1) for all 2 ≤ p ≤ ∞.
Remark 3.7. In Proposition 3.6 since V = V −1 it follows that k V is self-adjoint. Hence h V commutes with u V and u V is a self-adjoint partial isometry with u 2 V being the support projection of h V . We shall repeatedly make use of these observations without further reference.
Proposition 3.8. Let x ∈ C[Γ] with finite frequency support F ⊆ Γ. For V a symmetric neighbourhood basis of the identity of G we have for every 2 < p < ∞ Proof. For p = 2 the proof is (3.2) and for p = ∞ the statement is obvious as L(Γ) is a von Neumann subalgebra of L(G). So we assume that 2 < p < ∞. Let 2 < q < ∞ be the L 2 -conjugate of p determined by p −1 + q −1 = 2 −1 . Let ε > 0 and let y ∈ C[Γ] be such that y Lq ( Γ) ≤ 1 and We may consider a tail of the symmetric basis V so that each V ∈ V is relatively compact and so that each of the following families are disjoint: (1) sV, s ∈ F ; (2) sV, s ∈ F y . Moreover, we may assume that (3) s 1 V t 1 and s 2 V t 2 are disjoint whenever s 1 , s 2 ∈ F , t 1 , t 2 ∈ F y with s 1 t 1 = s 2 t 2 .
The frequency support of y * is F y . Since we assumed that sV, s ∈ F y are disjoint it follows from (3.1) -but then applied to y -that, . Using (2.3) and (3.4) we find for V ∈ V, We have By the Plancherel identity and disjointness assumption (3) in the first paragraph of this proof, .

Asymptotic intertwiners.
We will need to make use of the following proposition to prove our De Leeuw restriction theorems: both in the local linear and in the multilinear setting.
Proof. For 2 ≤ q < p < ∞ the proposition is exactly Claim B of [CPPR15]. The case 1 ≤ p < q ≤ 2 the same proof holds with trivial modifications with only one remark taken into account. Namely, at the place where [CPPR15, Corollary 1.4] was used (which is stated only for 2 ≤ q < ∞) we need to use [CPR18, Lemma 3.1] instead (which extends this corollary for 1 < q < 2). Note that [CPR18, Lemma 3.1] was only stated for finite von Neumann algebras, but its proof remains valid in the semi-finite setting exactly as it is stated.
Proof. For p = 2 the statement is obvious. By complex interpolation the logarithm of the norms on both sides of (3.8) is a convex function in p and hence certainly continuous. So the case p = 1 follows from the case 1 < p < 2 by approximation. The case 1 < p < 2 follows by duality from the case 2 < p < ∞, see Remark 2.2.
The frequency support F of T m| Γ (x) is finite and contained in supp(m| Γ ). Let ε > 0. Let V be a symmetric neighbourhood basis shrinking to the identity of G such that By Proposition 3.9 the last summand goes to 0. Hence using again Proposition 3.8, Letting ε ց 0 concludes the proof as C[Γ] is dense in L p ( Γ).
Remark 3.12. In Theorem 3.11 if moreover G ∈ [SAIN] Γ then δ F = 1 for any F ⊆ Γ finite and we recover [CPPR15, Theorem A]. In particular our theorem also generalizes the restriction theorem from [Lee65].
Remark 3.13. Use the notation of Theorem 3.11. Now let N be any (semi-finite) von Neumann algebra and consider the tensor product L p (N ) ⊗ L p ( G), which is equipped with the L p norm induced by the tensor product of the weights on N and L(G). We write L p ( G; L p (N )) for L p (N )⊗ L p ( G). Consider the vector-valued extension T N m of T m given by L p ( G; L p (N )) → L p ( G; L p (N )), Let R be the hyperfinite II 1 -factor. We will say that T m is completely bounded if T R m is bounded and write T m cb = T R m . We still have (3.8) in the cb-norm since we may apply the theorem Example 3.14. Consider G = SL(n, R) which has real rank n − 1. Set Γ = SL(n, Z). As a particular case of Theorem 8.1 proved below (see Remark 8.2) we obtain that we verify next as a sanity check. This shows that our methods provide an entry to construct multipliers on SL(n, Z).
In [Mau07, Corollary 1.1 and Example 4.2] it is shown that Indeed, let a be a maximally noncompact Cartan subalgebra of the Lie algebra g of G and let λ 1 , . . . , λ n ∈ a * be the natural weights described in [Mau07, Example 4.1, 4.2]. The weights occurring in the adjoint representation are λ i − λ j , 1 ≤ i, j ≤ n and C is defined as the convex hull of these. For β the sum of all positive roots (see [Mau07]; so not the usual half-sum) we find, If n is even β/D is on the face of C given by the convex hull of the weights If n is odd then β/D is on the face of C given by the convex hull of the weights So we have determined D and E as in [Mau07] and [Mau07, Corollary 1.1] yields (3.10). Note that D and E in this example are written as d and e in [Mau07] which differs from d as in Theorem B.
3.4. Asymptotic isometries for 1 ≤ p < 2 under the [SAIN] condition. In order to prove the multilinear theorems we prove two lemmas that in particular extend the results of Section 3.1 under the [SAIN] condition. Recall that the group C * -algebra C * r (Γ) is defined as the norm closure of C[Γ] in B(ℓ 2 (Γ)).
Proof. By approximation and taking linear combinations it suffices to prove the lemma in case x = λ(r) with r ∈ G. Then [CPPR15, Corollary 1.4] shows that for 2 ≤ p < ∞, . (3.12) Taking the limit over V ∈ V this converges to 0 by the [SAIN] condition, c.f. Remark 3.4. Now for any 1 ≤ p < ∞, . By Proposition 3.6 and (3.12) this expression converges to 0.
For completeness we note that the crucial inequality behind (3.12) is Ando's inequality [CPPR15, Lemma 2.2] from which it could also be proved directly using the methods in [CPPR15, Section 2].
and assume x = y * y. Then we have, by Lemma 3.15, Proposition 3.8 and similarly, by Lemma 3.15, .
So we may conclude the proof from the self-adjoint case. Now for general x ∈ C * r (Γ) we may replace x and h V by respectively, ) respectively and conclude the proof; it follows that the first part of the proof also holds in this 2 × 2-matrix amplification. One way to see this is using the fact that the group von Neumann algebra of the infinite permutation group S ∞ is the hyperfinite II 1 factor, R. We now replace the inclusion Γ < G by S ∞ × Γ < S ∞ × G. The corresponding von Neumann algebras are then R ⊗ L(Γ) and R ⊗ L(G). Since the matrix algebras embed into R, we may view the matrix amplification of x as an element of L p (R ⊗ L ∞ ( Γ)).

Theorem C: The multilinear De Leeuw restriction theorem
In this section we define multilinear Fourier multipliers and prove the multilinear De Leeuw restriction theorem. 4.1. Multilinear maps. Let X 1 , . . . , X n and Y be Banach spaces. Let T : X 1 × . . . × X n → Y be a multilinear map, meaning that it is linear in each of its variables. The norm of T is then defined as 4.2. Multilinear Fourier multipliers. Let G be a locally compact unimodular group. Let m : G ×n → C be a bounded measurable function. Consider the tuple (p 1 , . . . , p n ) with 1 ≤ p i < ∞ and let also 1 ≤ p < ∞. We say that m is a (p 1 , . . . , p n ) → p multiplier if there exists a bounded n-linear map for all f 1 , . . . , f n ∈ C c (G) * 2 . By mild abuse of terminology we also refer to the map T m as the (p 1 , . . . , p n ) → p multiplier and m is then explicitly referred to as the symbol of T m . If p is the conjugate determined by p −1 = n j=1 p −1 j then we simply speak about a (p 1 , . . . , p n )-multiplier in analogy with the linear case n = 1. n is also called the order of the multiplier. 4.3. Reduction formulae. We prove a number of lemmas that reduce the analysis of n-linear Fourier multipliers to Fourier multipliers of lower order.
For 1 ≤ k ≤ n let (q 1 , . . . , q k ), 1 ≤ q j < ∞ be a consummation of (p 1 , . . . , p n ), 1 ≤ p j < ∞ at indices 1 = i 1 < . . . < i k ≤ n. Suppose that m : G ×k → C is bounded and measurable and set Then m is a (q 1 , . . . , q k )-multiplier iff m is a (p 1 , . . . , p n )-multiplier and moreover, for Proof. Let f 1 , . . . f n ∈ C c (G) * 2 . Then, using the invariance of the Haar integral for the second equality and (2.1), So if T m is bounded it follows by this expression and the Hölder inequality that also T m is bounded.
and taking the supremum over all such f i we conclude the proof.
Then m is a (p 1 , . . . , p k ) → p-multiplier if and only if so is m i ( · ; r, t, r ′ ). In that case for Proof. The proof of (4.1) is similar to Lemma 4.2, namely one verifies the equality for x j = λ(f j ), f j ∈ C c (G) * 2 by an elementary computation and then concludes by continuity. Since multiplication with λ(s), s ∈ G determines an isometry on L p j ( G) that is strongly continuous in s by [JuSh05, Lemma 2.3] the final statements follow as well.
There are several formulae available for nested Fourier multipliers of which the following lemma is a particular case.
Then m is a (p 1 , . . . , p n )-multiplier and for Proof. Again the identity (4.2) can directly be verified for The boundedness of T m then follows as all T m j are bounded together with the Hölder inequality.

4.4.
Proof of Theorem C. The idea behind the following proof is that we approximate a multiplier by multipliers that decompose as a nested iteration of linear multipliers. Then using the analysis of linear multipliers we may prove the desired result. We first give the core of the proof of Theorem 4.5 and prove the intertwining property that is needed for it afterwards in Lemma 4.6.
Theorem 4.5 (Theorem C). Let G be a locally compact unimodular group. Let Γ < G be a discrete subgroup such that G ∈ [SAIN] Γ . Let m ∈ C b (G ×n ). Then for every 1 ≤ p, p 1 , . . . , p n < ∞ with p −1 = n j=1 p −1 j we have that Proof. In case n = 1 the theorem was proved in [CPPR15, Theorem A] or Remark 3.12. We therefore assume that n ≥ 2. Consequently 1 < p 1 , . . . , p n < ∞. We note however that our proof remains valid for n = 1 provided that 1 < p = p 1 < ∞. Let (ϕ k ) k be a net of positive definite functions in the Fourier algebra A(G) = L 2 (G) * L 2 (G) with ϕ k 1 = 1 and compact support shrinking to the identity of G. Recall that elements of A(G) are continuous and p-multipliers for all 1 ≤ p ≤ ∞ (see [CaHa85]). For any function (4.5) Since m k → m pointwise and all x j have finite frequency support we find that Let V be a symmetric neighbourhood basis of the identity of G as in Remark 3.4. By Proposition 3.16, Now by Lemma 4.6 -which we prove below -the limit over k and V in the second summand exists and is 0. Therefore, By Lemma 4.3 we find that T m k ≤ T m . Hence, together with (3.1) of Proposition 3.6, we conclude that x j Lp j ( Γ) .
Since the elements x j ∈ C[Γ] are dense in L p ( Γ) we conclude the proof.
Lemma 4.6. In the proof of Theorem 4.5, in particular with n ≥ 2, we have that If n = 1 the statement also holds in case 1 < p = p 1 < ∞.
Proof. Recall that all x j have finite frequency support and therefore by the triangle inequality it suffices to consider the case where x j = λ(r j ), r j ∈ Γ. Let ζ : G → R ≥0 be a compactly supported continuous, positive definite function in the Fourier algebra A(G) with ζ(e) = 1, so that T ζ is a contraction for all 1 ≤ p ≤ ∞. For 1 ≤ j ≤ n, s ∈ G let (4.6) ζ j (s) = ζ(r −1 j s).
As observed in the first paragraph the latter expression converges to 0 in k, so we find that lim k lim V ∈V A k,V = 0.
Next we treat the summand C k,V . Define, using the notation (4.4), Since by similar arguments to the ones used in Lemmas 4.2 and 4.4, we find We have see also Proposition 3.6. Recall (4.6) and the fact that ζ(e) = 1 and x j = λ(r j ). By Lemma 3.15 and using Proposition 3.9 we see that for every 1 ≤ j ≤ n − 2, Hence from (4.8), (4.9) Now going back to the original quantity we have to estimate, we find by Lemma 4.3, Taking limits in (4.9) yields, This expression goes to 0 when taking the limit in k. This concludes the estimate for C k,V . We now prove that for every k we have lim V B k,V = 0. For what follows, we will assume, by scaling ϕ k if necessary, that T ϕ k is a contraction on L p ( G); we may do this since at this point we need to prove convergence in V for fixed k and our arguments do not rely on the earlier assumption ϕ k 1 = 1 anymore. Set the function When we see ψ k for fixed t 1 , . . . , t n as a function of the variables s 1 , . . . , s n we shall write ψ k ( · ; t 1 , . . . , t n ). By the last expression of (4.5) we see that We justify that the integrand is dominated by an integrable function that does not depend on V . Indeed, note that T ψ k ( · ;t 1 ,...,tn) and T ψ k ( · ;t 1 ,...,tn)| Γ ×n are contractive (this follows from the expansions (4.11) and (4.12) below) and in combination with Proposition 3.6 it follows that the integrand on the second line of (4.10) is at most 2 for any t 1 , . . . , t n ∈ G. Since m(ζ 1 , . . . , ζ n ) is integrable, the integral (4.10) is majorized by G ×n 2|(m(ζ 1 , . . . , ζ n ))(t −1 1 t 2 , . . . , t −1 n−2 t n−1 , t −1 n )|dt 1 . . . t n = 2 m(ζ 1 , . . . , ζ n ) L 1 (G ×n ) < ∞.
So if we can show that the limit in V of the integrand of (4.10) converges to 0 pointwise then by the Lebesgue dominated convergence theorem we conclude that for all k we have lim V B k,V = 0. Introduce the short hand notation for the multipliers T j = T ϕ k ( · t j ) and S j = T ϕ k ( · t j )| Γ for 1 ≤ j ≤ n − 1. We also set T n = T ϕ k (tn · ) and S n = T ϕ k (tn · )| Γ For any y j ∈ L p j ( G) we have by Lemma 4.4, (4.11) T ψ k ( · ;t 1 ,...,tn) (y 1 , . . . , y n ) = T 1 (y 1 (T 2 (y 2 . . . T n−1 (y n−1 ) . . .)))T n (y n ).
Now fix y j := y j, for a while V shall be fixed and at the very end of the proof we will take a limit in V . Set 1 < q j < ∞ by q −1 j = n−1 i=j p −1 i . Note that q −1 1 + p −1 n = p −1 . Then set for 0 ≤ j ≤ n − 1, ) . . .))).

By definition
R n−1,V =T 1 (y 1 (T 2 (y 2 . . . T n−1 (y n−1 ) . . .))), and these expressions should be compared to (4.11) and (4.12). Also let R denote the term S 1 (x 1 (S 2 (x 2 . . . S n−1 (x n−1 ) . . .))). S j and T j are contractions and therefore we record at this Then -in view of (4.10) -we estimate using the triangle inequality V and that T n = T ϕ k ( · tn) is a multiplier both on L 1 and L ∞ . Therefore by the Hölder inequality and Proposition 3.9, It remains to show that the final summand in (4.13) goes to 0 in V . For 1 ≤ j ≤ n − 1, set We estimate, (4.14) Since T j is a contraction the first summand goes to 0 in V by Lemma 3.15. For the second summand we note that T j = T ϕ k ( · t j ) is a multiplier both on L 1 ( G) and L ∞ ( G). Since 1 < q j < ∞ we get that also the second summand in (4.14) goes to 0 by Proposition 3.9.
Remark 4.7. If in Lemma 4.6 we have x 1 = . . . = x n = 1 then the proof holds for any symmetric neighbourhood basis V not necessarily witnessing the [SAIN] Γ condition. Remark 4.8. As in Remark 3.13, we note that Theorem C also has a completely bounded version. Use the notation of Theorem 4.5 and let N be any (semi-finite) von Neumann algebra. The vectorvalued extension T N m of T m is given by L p 1 ( G; L p 1 (N ))×L p 2 ( G; ; L p 2 (N ))×. . .×L pn ( G; L pn (N )) → L p ( G; L p (N )), Let R be the hyperfinite II 1 -factor. As in the linear case, we will say that T m is completely bounded if T R m is bounded and write T m cb = T R m . Then we still have (4.3) in the cb-norm.

Multilinear lattice approximation
The aim of this section is to prove a multilinear version of [CPPR15, Theorem C].
Definition 5.1. A locally compact group G is approximable by discrete subgroups (notation G ∈ ADS) if there exists a sequence of discrete lattices (Γ j ) j∈N of G with associated fundamental domains X j which form a neigbourhood basis of the identity.
By passing to a subsequence we may assume that all X j in Definition 5.1 are relatively compact. In other words Γ j is cocompact. The following theorem shows that for ADS groups the norm of Fourier multipliers can be approximated by looking at the restriction of the symbol to appropriate lattices.
Theorem 5.2. Let G ∈ ADS with approximating cocompact lattices (Γ j ) j∈N . Let m ∈ C b (G ×n ) and let 1 ≤ p, p 1 , . . . , p n < ∞ be such that p −1 = n i=1 p −1 i . Then, Before proving the theorem we introduce the auxiliary maps from [CPPR15, Theorem C]. Let X j be shrinking relatively compact fundamental domains for the inclusions of Γ j in G. For s ∈ G we shall write γ j (s) for the unique element in Γ j such that s ∈ γ j (s)X j . Let h j = λ(1 X j ). For x ∈ C[Γ j ] we set the elements in L ∞ ( G) by The proof of [CPPR15, Theorem C, p. 19-20] shows that Φ Since x has compact frequency support these summations are finite. The proof of [CPPR15, Theorem C, p. 19-20] shows that Ψ (p) j extends to a contraction L p ( G) → L p ( Γ). Now set Recall that we defined φ ∨ (s) = φ(s −1 ) for φ a function on G. The proof of Theorem 5.2 hinges on the following lemma.
Hence we see that this term goes to 0 proving (5.3).
Proof of Theorem 5.2. Let x 1 , . . . , x n be as in Lemma 5.3 and assume x i Lp i ( G) ≤ 1. Lemma 5.3 shows (for p = 1 this requires Kaplansky's density theorem), Then, as Φ This concludes the proof.
We note that we may use the lattice approximation theorem to obtain a stronger version of the restriction theorem. The following follows directly by combining Theorem 4.5 and Theorem 5.2.
Theorem 5.4. Let H be a subgroup of a locally compact group G. Let H ∈ ADS and G ∈ [SAIN] H . Let m : G ×n → C be a bounded continuous symbol giving rise to a multilinear (p 1 , p 2 , . . . , p n )-multiplier on G with 1 ≤ p, p i < ∞ and p −1 = n i=1 p −1 i . Then

Other multilinear de Leeuw theorems: periodization and compactification
In this section we prove multilinear versions of the periodization and compactification theorem. The proofs in this section are very similar to the linear case in [CPPR15] or to other proofs in this paper. Therefore, we only give a sketch of the proofs.
Given a locally compact group G, let G disc denote the same group equipped with the discrete topology. The compactification theorem relates the (p 1 , p 2 , . . . , p n )-boundedness of a Fourier multiplier on G with the (p 1 , p 2 , . . . , p n )-boundedness of a Fourier multiplier on G disc . We recall that if G is abelian G disc is known as the Bohr compactification of G. We refer to the discussion in [CPPR15, Section 6] for why the hypotheses in the following theorem are the most natural ones.
Theorem 6.1. Let G be a locally compact group and let m ∈ C b (G ×n ). Let 1 < p, p 1 , . . . , p n < ∞, Proof. (i) Let (Γ j ) j∈N be a sequence of approximating cocompact lattices as in the hypothesis of Theorem 5.2. Then, Now Γ j < G disc is an inclusion of two discrete groups so that the natural inclusion C[Γ] ⊆ C[G disc ] extends to an isometric inclusion L p ( Γ j ) ⊆ L p ( G disc ). Therefore, T m| Γ ×n j is just a restriction of T m and hence (ii) This is essentially a restriction result for the subgroup G disc . As such, the proof is similar to Theorem 4.5 and we sketch its main difference here. From [CPPR15,Theorem 8.7] we know that the amenability of G disc implies that G ∈ [SAIN] G . Now for F ⊂ G finite let Γ F be the smallest (not necessarily closed) subgroup of G containing F . Then Γ F is countable and we may still perform the construction in Remark 3.4 with Γ replaced by Γ F . Hence we obtain a local neighbourhood basis V of the identity of G such that for every F ′ ⊆ Γ F finite we have lim V ∈V δ F ′ (V ) = 1. Then Proposition 3.6 still holds for any x ∈ C[ Γ F ] with V ∈ V sufficiently small.
By [CPPR15,Lemma 6.1] the amenability of G disc implies that for any x ∈ C[G disc ] we have x C * r (G disc ) = x C * r (G) . So naturally we have an isometric embedding C * r (G disc ) ⊆ L ∞ ( G). Consequently, the proof of Proposition 3.6 still holds, where for the p = ∞ case the latter fact is used. Since Γ F is a discrete subgroup of G disc we have C * r (Γ F ) ⊆ C * r (G disc ) which thus is included in L ∞ ( G). Then Lemma 3.15 and Proposition 3.16 hold for any x ∈ C * r (Γ F ). This suffices then to run the proofs of Theorem 4.5 and Lemma 4.6 by putting F as in the previous paragraph equal to the union of the frequency supports of x 1 , . . . , x n occurring in the proof; note that products of the x i 's then have frequency support contained in Γ F . The proof of the periodization theorem is a direct modification of the linear case (Theorem D (iii) and (iv) in [CPPR15]).
Theorem 6.2. Let G be a locally compact group and let H be a closed normal subgroup of G. Let m q ∈ C b ((G/H) ×n ) and let m π ∈ C b (G n ) denote its H-periodization m π (g 1 , . . . , g n ) = m (g 1 H, g 2 H, . . . , g n H). Then, the following inequalities hold for 1 < p, Proof. (i) was proved in the linear case by Saeki [Sae70] and in the multilinear case by [Rod13]. The proof of (ii) remains essentially unchanged from the linear case proved in [CPPR15,Section 7]. As in [CPPR15, Section 7] we note that the operator defined by is a central projection of L(G) such that λ(s)Π = Π = Πλ(s) for all s ∈ H. Moreover, this lets us define a normal * -homomorphism π preserves the Plancherel weight on L(G/H), so it induces an isometry π p : L p ( G/H) → L p ( G)Π for 1 ≤ p ≤ ∞. Since Π is a central projection in L(G), a similar computation as in [CPPR15] now shows us that π p • T mq (x 1 , . . . , x n ) = (G/H) n m q (g 1 H, g 2 H, . . . , g n H) x 1 (g 1 H) . . . x n (g n H)λ(g 1 . . . g n )Πdµ G/H (g 1 H) . . . dµ G/H (g n H) . This concludes (ii) since π p is an isometry for all 1 ≤ p ≤ ∞.

Example: Multilinear multipliers on the Heisenberg group
We provide an example of multilinear multipliers on a non-abelian group. Let be the (2 + 1)-dimensional Heisenberg group. We shall see H as R 3 with the group law H is unimodular and its Haar measure is just the usual Lebesgue measure of R 3 . There is an action R R 2 by x·(y, z) = (y, z +xy). Then H = R 2 ⋊R. Recall our definition of the completely bounded norm from Remark 4.8.
Set the unitary U = s∈Λ P s ⊗ λ(s) ∈ B(ℓ 2 (Λ)) ⊗ L ∞ ( Λ). Then set the trace preserving *homomorphism, Then π determines an isometry As described in Remark 3.13, L p ( Γ) ⊗ S p here refers to the tensor product equipped with the L p norm given by the tensor products of the weights on Γ and N. Since Γ is amenable we have a trace preserving embedding L(Γ) ⊆ R by Connes' theorem [Con76] where R is the hyperfinite II 1 factor. Also R ⊗ M k (C) embeds into R ⊗ R ≃ R in a trace preserving way. Thereforeapproximating S p with L p (M k (C)) -since T m is completely bounded we may extend T m to a vector valued map and we have T vec m • (π p 1 × π p 2 ) = π p • S M . Therefore S M is bounded by the norm of the right hand side of (7.1).
By the Følner condition the limit of the fraction is 1 and so this expression equals T M (x, y), z p,p ′ . Since j p,α are contractions (see [Gon18]), this shows (for p = 1 with Kaplansky's density theorem) that the norm of T M is bounded above by the norm of S M , which in turn is bounded by T vec m cb .
We have for 1 ≤ p, p 1 , j ∈ N, viewed as a cocompact discrete subgroup of H with ∪ j∈N H j dense in H. Set Γ j = j −1 Z×j −2 Z and Λ j = j −1 Z. The action R R 2 described above restricts to an action Λ j Γ j and H j = Γ j ⋊ Λ j . We now have by Theorem 5.2, Proposition 7.1 and Theorem 4.5 (see Remark 4.8), Remark 7.3. We note that the hypotheses of Theorem 7.2 are satisfied with finite norms in (7.3) if m obeys a suitable Hörmander-Mikhlin type condition, see [DLMV20a], [DLMV20b].
Remark 7.4. The methods in this section in fact work for more general semi-direct products of the form H = G ⋊ R n for an action by automorphisms of R n on a locally compact amenable group G provided that G is ADS and the approximating groups Γ j of G are invariant under the action of 1 j Z n ⊆ R n .

Theorem B: Lower bounds on δ for reductive Lie groups
In the following sections 8 through 10, we proceed with the proof of Theorem B. Let G be a real reductive Lie group with Lie algebra g, Cartan involution θ, maximal compact subgroup K, and invariant bilinear form B, cf. [Kna96, §VII.2]. The inner product B θ (x, y) := −B(x, θy) on g endows End(g) with the operator norm A → A . For ρ ≥ 1, we denote by (8.1) B G ρ := g ∈ G ; Ad g ≤ ρ the preimage under the adjoint representation Ad : G → End(g) of the closed ball of radius ρ around the origin. The aim of this section is to prove the following lower bound on δ B G ρ in terms of the radius.
Theorem 8.1 (Theorem B). If d is the maximal dimension of a nilpotent orbit of G in g, then As the adjoint orbit O X of a nilpotent element X ∈ g is a symplectic manifold, the coefficient d/2 in (8.2) is an integer. Since T X O X = g/g X is the quotient of g by the centralizer g X of X, the maximal dimension d can be expressed as where N ⊆ g is the nilpotent cone of g. In particular, d = 0 if g is compact or abelian, d = dim(g) − rk(g) if g is split (or quasisplit [Rot72, Thm 5.1]), and d = 2(dim C (g) − rk C (g)) if g is complex. In particular, d = n(n − 1) for SL(n, R) and GL(n, R), and d = 2n(n − 1) for SL(n, C) and GL(n, C).
Let g = k ⊕ p be the Cartan decomposition of g, let a be a maximal abelian subspace of p, and let A be the corresponding analytic subgroup. If g = k 1 ak 2 is the KAK-decomposition of g ∈ G ([Kna96, §VII.3]), then Ad g = Ad a since both B and θ are invariant under Ad(K). Let g = g 0 ⊕ λ∈Σ g λ be the restricted root space decomposition, where the sum runs over the set Σ ⊆ a * of restricted roots. Since A is simply connected, the set B G ρ can be equivalently described as (8.4) B G ρ = Kρ P K := K exp log(ρ)P K, where P ⊆ a is the polygon P = {h ∈ a ; α(h) ≤ 1 ∀ α ∈ Σ}. From this description, the following result easily follows. 8.1. A neighbourhood basis. The reductive Lie algebra g decomposes as the direct sum g = g 0 ⊕ z of the maximal semisimple ideal g 0 = [g, g] and the centre z. The former admits the Cartan decomposition g 0 = k 0 ⊕ p 0 , with k 0 = k ∩ g 0 and p 0 = p ∩ g 0 .
Let B z r ⊆ z and B g 0 r ⊆ g 0 be the open balls of radius r with respect to the inner product B θ in z and g 0 , respectively. Both are K-invariant, and the global Cartan decomposition G = K exp(p) implies that B z r ⊆ z is invariant under all of G. We are interested in the neighbourhood basis (8.5) V g ε,R,r := Ad G (B g 0 ε ) ∩ B g 0 R × B z r , obtained by intersecting the G-invariant neighbourhoods with the bounded sets B g 0 R × B z r . It will be convenient to write V g ε,R,r = V g 0 ε,R × B z r , where V g 0 ε,R is the bounded open subset of g 0 defined by Remark 8.4 (K-invariance). Since B g 0 ε is K-invariant, the global Cartan decomposition G = K exp(p) yields Ad G (B g 0 ε ) = Ad exp(p) (B g 0 ε ). Further, since z acts trivially on g 0 , we have Ad G (B g 0 ε ) = Ad exp(p 0 ) (B g 0 ε ) for p 0 = p ∩ g 0 . It follows that V g ε,R,r = V g 0 ε,R × B z r is the product of the K-invariant set V g 0 ε,R ⊆ g 0 that depends only on the restriction of B θ to g 0 , and the G-invariant set B z r ⊆ z that depends only on the restriction of B θ to z. In particular the sets V g ε,R,r ⊆ g are K-invariant, and they depend only on the triple (g, θ, B), not on the Lie group G.
The lower bound (8.2) will be established by calculating where µ is a Haar measure on G. Since V g ε,R,r = −V g ε,R,r , the sets exp(V g ε,R,r ) constitute a symmetric neighbourhood basis of the identity. It follows that , so in order to establish (8.2), it suffices to prove that δ 0 8.2. Relation between Haar measure and Lebesgue measure. The first step is to reformulate this in terms of the Lebesgue measure on the Lie algebra g. Let Vol G be a left invariant volume form on G, so that integrating against Vol G corresponds to a left Haar measure µ on G. Let Vol g be a constant volume form on g, corresponding to a Lebesgue measure Λ on g. We normalize these volume forms in such a way that Vol g agrees with exp * Vol G at the origin in g. Then exp * Vol G = νVol g , where the density ν of exp * Vol G with respect to the Lebesgue measure satisfies ν(0) = 1. We show that ν can be chosen arbitrarily close to 1 on U ε ⊆ g for small ε.
Proposition 8.5. The density is given by ν(x) = det(Φ x ), where Φ : g → End(g) is the left logarithmic derivative of the exponential map. The function ν : g → R is smooth, G-invariant, and equal to 1 on z ⊆ g. Moreover, there exists a constant c g 0 > 0 (depending only on g 0 ) such that for ε sufficiently small, ν − 1 ∞ ≤ c g 0 ε uniformly on U ε .
Proof. Let Φ : g → End(g) be the left logarithmic derivative of the exponential map, defined by Φ x (y) : Then for all x, y 1 , . . . , y n ∈ g, where the last two steps use that Vol G is left invariant, and that exp * Vol G agrees with Vol g at the origin. It follows that ν(x) = det(Φ x ). Since exp : g → G is equivariant with respect to the adjoint action on g and the conjugate action on G, its logarithmic derivative satisfies Φ Ad g (x) = Ad g • Φ x • Ad g −1 . In particular, ν(x) = det(Φ x ) is invariant under the adjoint action.
In fact, the left logarithmic derivative Φ x : g → g of the exponential map is explicitly given (cf. [Fa08, Thm. 2.1.4]) by the convergent power series Note that the determinant of the real linear map Φ x : g → g is equal to the determinant over C of the complexification Φ C x : g C → g C . In terms of the Jordan-Chevalley decomposition ad x = ad xs + ad xn into a semisimple and a nilpotent element of g C , we thus find where µ i are the eigenvalues of ad x as a complex linear transformation of g C . In particular, ν(x) depends only on the second factor in g = z ⊕ g 0 , and we can write ν(x) = ν 0 (ad x ) for a smooth function ν 0 : g 0 → R with ν 0 (0) = 1. It follows that for any c g 0 > ∇ν 0 (0) , there exists an ε 0 > 0 such that ν 0 − 1 ≤ c g 0 ε uniformly on B g 0 ε for all ε < ε 0 . The uniform estimate for ν on U ε now follows from G-invariance.
Since exp is equivariant under the adjoint action on g and G, this allows us to express δ 0 B G ρ in terms of the Lebesgue measure on g, .
Proposition 8.6. The number δ 0 B G ρ depends on G only through the maximal semisimple ideal , and the restriction of B θ to g 0 .
Proof. By Remark 8.4 the sets V g ε,R,r are K-invariant. If g = k exp(p) is the global Cartan decomposition of g ∈ G = K exp(p), we thus have Further, since z acts trivially on g, the decomposition p = p 0 + p z of p ∈ p with respect to p = p 0 ⊕ (z ∩ p) yields Since B G ρ is K-invariant (Proposition 8.3), we have k exp(p) ∈ B G ρ if and only if exp(p) ∈ B G ρ , which is the case if and only if exp(p 0 ) ∈ B G ρ . But since Ad exp(p 0 ) acts by the identity on second factor of g = g 0 ⊕ z, we have Ad g exp(p 0 ) ≤ ρ for the adjoint action on g if and only Ad g 0 exp(p 0 ) ≤ ρ for the adjoint action on g 0 . If G 0 denotes the connected adjoint group of the semisimple Lie algebra g 0 , we thus have where Λ g and Λ g 0 denote the Lebesgue measure on g and g 0 , respectively.
8.3. Proof of Theorem 8.1. By Proposition 8.6, it suffices to prove Theorem 8.1 for the case where G is the adjoint group of the semisimple Lie algebra g 0 . The proof hinges on the following lemma.
Lemma 8.7 (Key Lemma). Let G be a connected, real reductive Lie group with semisimple Lie algebra g. Then for all R > 0 and all ρ > 1, we have The proof of this lemma requires a rather detailed discussion of limits of orbital integrals, and will be deferred to Section 10. Assuming Lemma 8.7, the proof of Theorem 8.1 is quite straightforward.
Proof of Theorem 8.1, assuming Lemma 8.7. In view of (8.3), the maximal dimension d of the nilpotent orbits is the same in g and g 0 . By Proposition 8.6, we may therefore assume without loss of generality that G is a connected, real reductive Lie group with semisimple Lie algebra g.
Since Ad g ≤ ρ, we have Ad The strategy to prove Lemma 8.7 is as follows. Since the closure of an adjoint orbit O x through x ∈ g contains 0 if and only if x is nilpotent, the set ε>0 V ε,R is the intersection of the nilpotent cone N with the unit ball B g R (0). The union of the nilpotent orbits O X of maximal dimension is a dense open subset of the nilpotent cone. Using results of Harish-Chandra and Barbasch-Vogan on limiting orbit integrals, we will show that as ε approaches 0, the volume of V ε,R with respect to the Lebesgue measure on g scales with R in the same way as the Liouville volume of the symplectic manifold O X ∩ B g R (0). Since the Kostant-Kirillov-Souriau symplectic form ω KKS O X on the cone O X scales as R under dilation, the corresponding Liouville volume form Vol KKS O X scales as R d/2 , yielding the factor ρ d/2 in Lemma 8.7.

Limits of orbital measures
In the remainder of this section, we focus on the proof of Lemma 8.7. From now on, we assume that the Lie algebra g is semisimple, and that the invariant bilinear form B is the Killing form κ. We will generally denote generic elements of g by x, y, z, elements of a Cartan subalgebra h ⊂ g by h, and elements of the nilpotent cone N ⊂ g by X, Y, Z. For a subset A ⊆ g, we denote the centralizer and the normalizer in g by Z g (A) and N g (A) respectively. On the group level, we similarly define Z G (A) := {g ∈ G : Ad g y = y ∀y ∈ A}, N G (A) := {g ∈ G : Ad g A ⊂ A}.
9.1. Regular elements of Lie algebras. We recall the notion of regularity in g. For x ∈ g consider the characteristic polynomial Let k(x) be the minimal index so that a k(x) (x) = 0. If k(x) = min y∈g k(y), we say x is regular. If A ⊂ g is any subset of g, we define A reg to be the set of regular elements in A. If h is a Cartan subalgebra (CSA) and C is a connected component of h reg , we call C an open Weyl chamber of h.
Remark 9.1. Our notion of regularity is different from a common definition where x is regular if its centralizer has minimal dimension among the centralizers of all x ′ ∈ g. Our current definition is used, for example, in [Bou05].
We will use the following standard properties of regular elements: Lemma 9.2. Let g be a real Lie algebra.
i) Regular elements in g lie in a unique CSA given by their centralizer. ii) If a single element in an adjoint orbit O x ⊂ g is regular, then all elements are.
iii) The set of regular elements is dense and open in g, and its complement in g is a Lebesgue null set. Vol Ox defines a Radon measure µ Ox on g. In particular, it is finite on compact subsets of g. On nilpotent orbits, these measures are homogeneous: Lemma 9.3. Let A ⊂ g be a Borel set, let X ∈ g a nilpotent element, and let k = dim O X /2. Then for all ρ > 0, we have Proof. By the Jacobson-Morozov theorem, the nilpotent orbit O X is a cone, i.e. ρ · O X = O X for all ρ > 0. Thus we have Denote by m ρ : O X → O X the multiplication by ρ. By definition of the KKS form we have for all X ′ ∈ O X and y, z ∈ g: as required.
The measure µ Ox on the adjoint orbit O x through x ∈ g yields the distribution D Ox on g defined by is an oriented inner product space of even dimension 2k, then the Pfaffian of a skew-symmetric linear map A : where Vol ∈ ∧ 2k V * is the volume form associated to the inner product B on the oriented vector space V , and ω A (v, w) = B(v, Aw) is the 2-form associated to A with respect to the inner product B. With V = h ⊥ and A = θ • ad h , this yields Recall that the Pfaffian is related to the determinant by Since the determinant is the product of the eigenvalues over C, we can determine |π(h)| from the eigenvalues of ad h on the complexification (h ⊥ ) C ∼ = g C /h C . In view of the root space decomposi- where the first map is the natural embedding and the second map the projection of the direct sum (9.4) onto the first direct summand. Since the Slodowy slice intersects O x transversally, the composition of these two maps is surjective. Using T x O x ∼ = ad x g and T X O X ∼ = Ad X g, this surjective map induces the following exact sequence: We obtain a canonical volume form on O x ∩ S X as the quotient of the KKS volume forms on O x and O X , which in turn gives rise to the measure m x,X . In [Har12, Chapter 2], the following limit of orbits is defined for all x ∈ g: One may think of this set as the limit of the orbits O εx as ε approches zero, hence as the orbits approach the nilpotent cone. Let us first show that every nilpotent orbit arises, in this sense, as a limit of regular orbits: Lemma 9.5. Every nilpotent orbit O X lies in the set N x for some regular x ∈ g.
Proof. The idea of this proof is essentially due to [BaVo80,p.48]. If X is nilpotent, choose an sl 2 -triple {X, Y, H} with X as the nilpositive and H as the semisimple element. Consider the associated Slodowy slice S X = X + Z g (Y ). Since G · S X is an open neighbourhood of O X , and since the set of regular elements is dense in g, there exists a regular element x ∈ G · S X . In other words, there exist g ∈ G and V ∈ Z g (Y ) with The centralizer Z g (Y ) is stable under ad H , hence we can decompose it into the eigenspaces of ad H on this space. Due to the structure theory of finite-dimensional sl 2 -modules (cf. [Bou05, Sec VIII.2]), the eigenvalues λ on these eigenspaces are all nonpositive: Consider then the element g t := exp − 1 2 log(t)H ∈ G. Then we have But this means that X ∈ N ∩ t>0 Ad G (tx) = N x .
Remark 9.6. A closer inspection of such orbital limits is given in [FrMe]. Their definition of N x coincides with the one given here by [FrMe,Remark 3.6].
We will need the following asymptotic expression for the orbital integrals. It is proven in [Har12, Cor 2.3], relying on [BaVo80, Theorem 3.2].
Definition 9.7. For x ∈ g, we define where the minimum is taken over all adjoint orbits O X contained in N x .
Theorem 9.8. Let g be a real, reductive Lie algebra, and let x ∈ g. Then for all f ∈ C ∞ c (g) we have where the sum is taken over the set of nilpotent orbits O X ⊂ N x which are of maximal dimension among all orbits contained in N x , and the volume of S X ∩ O x is calculated with respect to the measure m x,X .
Remark 9.9. If C is an open Weyl chamber of some CSA h, one actually has N h = N h ′ for all h, h ′ ∈ C by [Har12, Cor 2.4] (originally attributed to [BaVo80]). In particular, the number m in Theorem 9.8 is equal for all h in one such open Weyl chamber. In this case we also write m(C) for the m associated to any h ∈ C.
9.4. Uniform Orbital Limits. Using results of Harish-Chandra and Varadarajan, we will show that the convergence in Theorem 9.8 is uniform on certain subsets of h. For an open subset C ⊆ V of a vector space V , we denote by C k (C, R) the space of functions u : C → R that are k times continuously differentiable on C, and whose derivatives of order at most k extend continuously to the closure. The following result follows from [HC57, Thm 3] (see also [Var77,Thm I.3.23]).
Theorem 9.10. Let h ⊂ g be a Cartan subalgebra, C ⊂ h an open Weyl chamber, and let f ∈ C ∞ c (g, R). Then the function u f : C → R defined by Remark 9.11. In fact, u f extends to a Schwartz function on the connected component of the non-zero sets of the singular imaginary roots.
Remark 9.12. The definition of the invariant integral in [HC57] is, up to a positive scalar, equivalent to ours by Proposition 9.4.  Fix again an open Weyl chamber C of some Cartan subalgebra h ⊂ g. From Theorem 9.8, Theorem 9.10, and the fact that integrals over compact domains commute with uniform limits, it immediately follows that for all compact K ⊂ h, all f ∈ C ∞ c (g) and all continuous functions w : h → R. In particular, the integral on the right-hand side is well-defined. A similar statement holds with D O X (f ) replaced by µ O X (B R (0)).
Corollary 9.14. Let h ⊂ g be a Cartan subalgebra, K ⊂ h a compact subset, and C ⊂ h an open Weyl chamber. Let w : h → R be a continuous function, and let R > 0. Then with the sum over the O X as in Theorem 9.8, and the right hand side is integrable.
Let k = dim O X /2 for any of the orbits O X in the sum. Since w can be written as the difference of two nonnegative functions, we may assume without loss of generality that w is nonnegative. Then, for all n ∈ N, we have (9.8) Note that µ O X (B R (0)) = µ O X (B R (0)) for all nilpotent elements X. Indeed, since O X is a cone, the boundary ∂B R (0) intersects O X transversally, and the intersection O X ∩ ∂B R (0) ⊂ O X is either empty (when X = 0) or a submanifold of codimension at least one (when X = 0). Its Liouville measure is thus zero by Sard's Theorem. Finally, the statement follows by taking the monotone limits n → ∞ in (9.8).
10. Proof of the key Lemma 8.7 concluding Theorem B In all that follows, we will fix a maximal set of mutually nonconjugate, θ-stable CSAs h 1 , . . . , h n of g, with associated Cartan subgroups H 1 , . . . , H n ⊂ G defined by H i := Z G (h i ). We denote by W j := N G (h j )/H j the Weyl group associated to h j . We will need the following Lie algebraic version of the well-known Harish-Chandra integral formula [HC65, Lemma 41], see e.g. [Var77, Part I, Section 3, Lemma 2].
Lemma 10.1. Let g be a reductive Lie algebra with connected adjoint group G, and let f ∈ L 1 (g, Λ g ) be an integrable function on g. Then, for j = 1, . . . , n, there are G-invariant volume forms Vol G/H j on G/H j , so that Here, |π j | is a product of positive roots of (g C , (h j ) C ) as in Proposition 9.4. Proof. To see that the right hand side is indeed a disjoint union, suppose that Ad g h = Ad g ′ h ′ for regular elements h ∈ h i , h ′ ∈ h j and g, g ′ ∈ G. Since h and h ′ are regular, h i = Z g (h) and h j = Z g (h ′ ). Since h is conjugate to h ′ , h i is conjugate to h j . As the various CSA's are mutually nonconjugate, we conclude that h i = h j . ⊂: Let Ad g x ∈ (V ε,R ) reg with g ∈ G and x ∈ B ε (0). Since x is regular, it lies in a unique CSA, and is conjugate to an element h ∈ h j for some j, i.e. ∃g ′ ∈ G : Ad g ′ x = h ∈ h j . By Lemma 9.2, the orbit of a regular element consists only of regular elements. Hence h ∈ (V h j ε ) reg and Ad g(g ′ ) −1 h = Ad g x ∈ B R (0). Hence Ad g x lies in the right hand side. ⊃: Let Ad g h lie in the right hand side. Because h ∈ Ad G B ε (0), we can write h = Ad g ′ x for some g ′ ∈ G and x ∈ B ε (0). But then Ad g h = Ad gg ′ x ∈ Ad G B ε (0), and since h was regular, so is every element in its orbit, and we have Ad g h ∈ (V ε,R ) reg .
It follows that ad * h = − ad θh . Since [h, θh] = 0, the operator ad h : g → g is normal, and its operator norm ad h with respect to the inner product κ θ satisfies ad h = max λ∈Spec(ad h ) |λ| .
Let g ∈ G such that Ad g (h) ∈ B ε (0). Since ad Adg h is conjugate to ad h , it is a normal operator with the same eigenvalues, so in particular ad h = ad Adg h . So since the operator norm is bounded on B ε (0), it is bounded on Ad G (B ε (0)) ∩ h as well, and the latter is a bounded subset of h.
Recall that V ε,R differs from (V ε,R ) reg only in a Lebesgue null set. Also, we can write every (h j ) reg as the union of its open Weyl chambers C j,r . Hence, we can use Lemma 10.1 and the decomposition To simplify, we will fix a single CSA h j and a single open Weyl chamber C j,r , and suppress the indices: H := H j , π := π j , h := h j , Vol G/H := Vol G/H j , C := C j,r . For now, let us look at the single summand of (10.2) corresponding to h and C.
Lemma 10.4. Fix the notation as in (10.3), let m := m(h) as in Definition 9.7 for an arbitrary h ∈ C, and R > 0 arbitrary. Then there is some c = 0 which does not depend on R so that where the sum is taken over all nilpotent orbits O X contained in N h for an arbitrary h ∈ C, cf. Remark 9.9. Proof. Recall the notation µ O h for the measure defined in Section 9, and that π(h) was defined in Equation In the last step, we used that V h ε = εV h 1 , that dΛ h (εh) = ε dim h dΛ h (h), and that π(εh) = ε |∆ + | π(h). By Corollary 9.14 and compactness of V h 1 ∩ C (Lemma 10.3), we have: This shows the statement.
Finally, we use this to prove the key lemma: Proof of key Lemma 8.7. Let O X ⊂ g be a nilpotent orbit of dimension d. By Lemma 9.5 and Remark 9.9, there is some CSA h ⊂ g and some open Weyl chamber C ⊂ h reg so that O X ⊂ N h for all h ∈ C. Since O X is of maximal dimension, the number m := m(x) from Definition 9.7 is minimal among m(x ′ ) for all x ′ ∈ g reg . Then, by (10.2) and Lemma 10.4, there are numbers c j,r = 0, independent of R, such that where the sum over the O X is carried out over all nilpotent orbits O X ⊂ N h of dimension d. Note that this sum is independent of the choice of h ∈ C j,r for fixed j and r, cf. Remark 9.9. All summands in the above sum are positive as they arise from volumes of subsets of g, and thus the total sum is nonzero. Lastly, the sum is homogeneous of degree d/2 in R by Lemma 9.3. Hence we may consider their quotient and conclude the proof: